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Resolución de integrales/ 5-x^2/ (5-x^2)^(-1)

Integral de (5-x^2)^(-1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1          
  /          
 |           
 |    1      
 |  ------ dx
 |       2   
 |  5 - x    
 |           
/            
0
∫0115−x2 dx\int\limits_{0}^{1} \frac{1}{5 - x^{2}}\, dx 
Integral(1/(5 - x^2), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-1, c=5, context=1/(5 - x**2), symbol=x), False), (ArccothRule(a=1, b=-1, c=5, context=1/(5 - x**2), symbol=x), x**2 > 5), (ArctanhRule(a=1, b=-1, c=5, context=1/(5 - x**2), symbol=x), x**2 < 5)], context=1/(5 - x**2), symbol=x)

  1. Añadimos la constante de integración:

    {5acoth⁡(5x5)5for x2>55atanh⁡(5x5)5for x2<5+constant\begin{cases} \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{\sqrt{5} x}{5} \right)}}{5} & \text{for}\: x^{2} > 5 \\\frac{\sqrt{5} \operatorname{atanh}{\left(\frac{\sqrt{5} x}{5} \right)}}{5} & \text{for}\: x^{2} < 5 \end{cases}+ \mathrm{constant}

Respuesta:

{5acoth⁡(5x5)5for x2>55atanh⁡(5x5)5for x2<5+constant\begin{cases} \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{\sqrt{5} x}{5} \right)}}{5} & \text{for}\: x^{2} > 5 \\\frac{\sqrt{5} \operatorname{atanh}{\left(\frac{\sqrt{5} x}{5} \right)}}{5} & \text{for}\: x^{2} < 5 \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                   //           /    ___\            \
                   ||  ___      |x*\/ 5 |            |
                   ||\/ 5 *acoth|-------|            |
  /                ||           \   5   /       2    |
 |                 ||--------------------  for x  > 5|
 |   1             ||         5                      |
 | ------ dx = C + |<                                |
 |      2          ||           /    ___\            |
 | 5 - x           ||  ___      |x*\/ 5 |            |
 |                 ||\/ 5 *atanh|-------|            |
/                  ||           \   5   /       2    |
                   ||--------------------  for x  < 5|
                   \\         5                      /
∫15−x2 dx=C+{5acoth⁡(5x5)5for x2>55atanh⁡(5x5)5for x2<5\int \frac{1}{5 - x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{\sqrt{5} x}{5} \right)}}{5} & \text{for}\: x^{2} > 5 \\\frac{\sqrt{5} \operatorname{atanh}{\left(\frac{\sqrt{5} x}{5} \right)}}{5} & \text{for}\: x^{2} < 5 \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.150.30
Trazar el gráfico f(x)
Respuesta [src] 
    ___ /          /       ___\\     ___    /  ___\     ___ /          /  ___\\     ___    /      ___\
  \/ 5 *\pi*I + log\-1 + \/ 5 //   \/ 5 *log\\/ 5 /   \/ 5 *\pi*I + log\\/ 5 //   \/ 5 *log\1 + \/ 5 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
                10                        10                      10                       10
−5log⁡(5)10+5log⁡(1+5)10−5(log⁡(−1+5)+iπ)10+5(log⁡(5)+iπ)10- \frac{\sqrt{5} \log{\left(\sqrt{5} \right)}}{10} + \frac{\sqrt{5} \log{\left(1 + \sqrt{5} \right)}}{10} - \frac{\sqrt{5} \left(\log{\left(-1 + \sqrt{5} \right)} + i \pi\right)}{10} + \frac{\sqrt{5} \left(\log{\left(\sqrt{5} \right)} + i \pi\right)}{10} 
=
= 
    ___ /          /       ___\\     ___    /  ___\     ___ /          /  ___\\     ___    /      ___\
  \/ 5 *\pi*I + log\-1 + \/ 5 //   \/ 5 *log\\/ 5 /   \/ 5 *\pi*I + log\\/ 5 //   \/ 5 *log\1 + \/ 5 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
                10                        10                      10                       10
−5log⁡(5)10+5log⁡(1+5)10−5(log⁡(−1+5)+iπ)10+5(log⁡(5)+iπ)10- \frac{\sqrt{5} \log{\left(\sqrt{5} \right)}}{10} + \frac{\sqrt{5} \log{\left(1 + \sqrt{5} \right)}}{10} - \frac{\sqrt{5} \left(\log{\left(-1 + \sqrt{5} \right)} + i \pi\right)}{10} + \frac{\sqrt{5} \left(\log{\left(\sqrt{5} \right)} + i \pi\right)}{10} 
-sqrt(5)*(pi*i + log(-1 + sqrt(5)))/10 - sqrt(5)*log(sqrt(5))/10 + sqrt(5)*(pi*i + log(sqrt(5)))/10 + sqrt(5)*log(1 + sqrt(5))/10
Respuesta numérica [src] 
0.215204470482002
0.215204470482002

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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